Copyright

Hyperbolic Functions: Properties & Applications

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Increasing Function: Definition & Example

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:02 Hyperbolic Functions…
  • 0:49 Identifying Hyperbolic…
  • 2:41 Examples of Hyperbolic…
  • 4:47 Other Uses of…
  • 5:30 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Timeline
Autoplay
Autoplay
Speed Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Although related to trigonometric functions, hyperbolic functions have special properties. In this lesson, you'll explore the properties of hyperbolic functions and their usage in both theoretical and real-world applications.

Hyperbolic Functions in Real Life

You and your friend had planned to walk to the movies together, but now it's five minutes to show time and no sign of him. Suddenly there he is, running on the sidewalk in front of your house, and it looks like stopping for you is not on his agenda. You jump off your porch and start to run. Heading towards your friend, you constantly change direction until you catch up.

Looking back at your footsteps, your friend remarks, 'Hey, you traced out an interesting curve.' To which you respond, 'For sure. It's based on hyperbolic functions.' Okay, okay, you probably wouldn't have said this. That is, not until you've completed this lesson.

Hyperbolic functions occur in all sorts of places. Let's talk about their properties and some of their cool applications.

Identifying Hyperbolic Functions

Hyperbolic functions are defined in terms of exponentials, and the definitions lead to properties such as differentiation of hyperbolic functions and their expansion as infinite series. They're written like the trig functions cosine (cos), sine (sin), tangent (tan), but they have an 'h' at the end. Cosh is pronounced 'kosh;' sinh is pronounced 'sinch;' and tanh is usually read as 'tan h' but sometimes we say 'tanch.'

Let's look at the graphs of cosh(x) and sinh(x):


null


That curve you traced is called the pursuit curve or the tractrix curve. The equations relating the x and y positions as a function of time both involve hyperbolic functions. In the following graph, do you see the square (you) pursuing the circle (your friend)?


null


Note the hyperbolic functions cosh(x) and tanh(x).

Identifying Hyperbolic Functions

The cosh(x) and sinh(x) functions may be defined in terms of exponential functions. Note that x is the independent variable.


null


null


Similarities exist between trig functions and hyperbolic functions. For example, tangent is sine divided by cosine. For hyperbolic tangent,


null


Trig functions have special names for their reciprocals. Secant, abbreviated sec, is one over cosine; cosecant, abbreviated csc, is one over sine; and cotangent, abbreviated cot, is one over tangent. The hyperbolic versions of these functions have an 'h' added to the end. Thus, we have sech(x), csch(x), and coth(x), as you can see:


null


null


null


Here are some other hyperbolic function properties.


null


In terms of their derivatives,


null


null


As a series,


null


null


Examples of Hyperbolic Functions

Let's use hyperbolic function properties.

Example 1

When a flexible cable is suspended between two towers, the relevant equation is:


null


We would like to show you that y = cosh(x) is a solution to this equation.

To solve this problem we'll start by assuming that y = cosh(x) is the solution.

Recall that the first derivative is called y prime and the second derivative is called y double-prime.

Starting with our assumption that y is cosh(x), y prime is the derivative of cosh(x). From the properties, the derivative of cosh(x) is sinh(x). Thus, y prime is sinh(x).

Continuing, y double-prime is the derivative of y prime. That means that y double-prime is the derivative of sinh(x). From the properties, we know that the derivative of sinh(x) is cosh(x). Thus, y double-prime is cosh(x).

You can see what we have so far:


null


Now, let's substitute into our equation. We get the following:


null


If we square both sides, we then arrive at the following:


null


Now, through some simple algebra we get this equation


null


which is one of our properties.

To unlock this lesson you must be a Study.com Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account
Support